Serre's reduction of linear functional systems

Serre's reduction aims at reducing the number of unknowns and equations of a linear functional system (e.g., system of partial differential equations, system of differential time-delay equations, system of difference equations). Finding an equivalent representation of a linear functional system containing fewer equations and fewer unknowns generally simplifies the study of its structural properties, its closed-form integration as well as of different numerical analysis issues. The purpose of this paper is to present a constructive approach to Serre's reduction for determined and underdetermined linear functional systems

On the Decoupling of a Class of Linear Functional Systems

Abstract Multivariate polynomial matrices arise from the treatment of functional linear systems such as systems described by partial differential equations, delay-differential equations or linear multidimensional discrete equations. In this paper we present conditions under which a class of multivariate polynomial matrices is equivalent to a block diagonal form. The conditions correspond to the decomposition of the associated linear systems of functional equations. The constructive method which can be easily implemented on a computer algebra system is illustrated by an example. Mathematics Subject Classification: 93C05, 93B11, 34C20, 68W30, 13P0

Reduction of Linear Functional Systems using Fuhrmann's Equivalence

Functional systems arise in the treatment of systems of partial differential equations, delay-differential equations, multidimensional equations, etc. The problem of reducing a linear functional system to a system containing fewer equations and unknowns was first studied by Serre. Finding an equivalent presentation of a linear functional system containing fewer equations and fewer unknowns can generally simplify both the study of the structural properties of the linear functional system and of different numerical analysis issues, and it can sometimes help in solving the linear functional system. In this paper, Fuhrmann's equivalence is used to present a constructive result on the reduction of under-determined linear functional systems to a single equation involving a single unknown. This equivalence transformation has been studied by a number of authors and has been shown to play an important role in the theory of linear functional systems

Strict System Equivalence of 2D Linear Discrete State Space Models

The connection between the polynomial matrix descriptions (PMDs) of the well-known regular and singular 2D linear discrete state space models is considered. It is shown that the transformation of strict system equivalence in the sense of Fuhrmann provides the basis for this connection. The exact form of the transformation is established for both the regular and singular cases

A New Approach to the Stabilization of the Wave Equation with Boundary Damping Control

This paper deals with boundary feedback stabilization of a system, which consists of a wave equation in a bounded domain of , with Neumann boundary conditions. To stabilize the system, we propose a boundary feedback law involving only a damping term. Then using a new energy function, we show that the solutions of the system asymptotically converge to a stationary position, which depends on the initial data. Similar results were announced without proof in (Chentouf and Boudellioua, 2004)

Characterization of a class of spatially interconnected systems (ladder circuits) using two-dimensional systems theory

This paper considers a class of spatially interconnected systems formed by ladder circuits using two-dimensional systems theory. The individual circuits in this class are described by hybrid (continuous/discrete) linear differential/difference equations in time (continuous) and spatial (discrete) variables and therefore have a two-dimensional systems structure. This paper shows that a ladder circuit model and models for 2-D dynamics have a well defined equivalence property and hence analysis tools can be transferred between them. Also the mechanism for transforming one to the other is established.</p

Constructing the singular Roesser state-space model description of 3D spatio-temporal dynamics from the polynomial system matrix

This paper considers systems theoretic properties of linear systems defined in terms of spatial and temporal indeterminates. These include physical applications where one of the indeterminates is of finite duration. In some cases, a singular Roesser state-space model representation of the dynamics has found use in characterizing systems theoretic properties. The representation of the dynamics of many linear systems is obtained in terms of transform variables and a polynomial system matrix representation. This paper develops a direct method for constructing the singular Roesser state-space realization from the system matrix description for 3D systems such that relevant properties are retained. Since this method developed relies on basic linear algebra operations, it may be highly effective from the computational standpoint. In particular, spatially interconnected systems of the form of the ladder circuits are considered as the example. This application confirms the usefulness and effectiveness of the proposed method independently of the system spatial order